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Strong convergence of the Euler scheme for singular kinetic SDEs driven by $α$-stable processes

Chengcheng Ling

TL;DR

The paper studies strong convergence of an Euler-type scheme for singular second-order SDEs driven by α-stable noise with degenerate coordinates. By leveraging regularization-by-noise techniques, a stochastic sewing framework, and sharp semigroup estimates in anisotropic Besov spaces, it proves existence/uniqueness of the solution and a strong convergence rate for the Euler approximation with rate n^{-(1/2+β/(α(1+α))∧1/2)+ε} for any ε>0, under anisotropic Hölder drift with β>1−α/2. An almost-sure convergence result is also established via a random dominating variable, with a rate matching the L_p bound up to ε. The analysis harmonizes second-order kinetic SDE behavior with first-order BDG-Lévy-type results, extending regularization-by-noise methods to this degenerate, jump-driven setting and providing rigorous guidance for numerical simulations of such systems.

Abstract

We study the strong approximation of the solutions to singular stochastic kinetic equations (also referred to as second-order SDEs) driven by $α$-stable processes, using an Euler-type scheme inspired by [11]. For these equations, the stability index $α$ lies in the range $(1,2)$, and the drift term exhibits anisotropic $β$-Hölder continuity with $β>1 - \fracα{2}$. We establish a convergence rate of $(\frac{1}{2} + \fracβ{α(1+α)} \wedge \frac{1}{2})$, which aligns with the results in [4] concerning first-order SDEs.

Strong convergence of the Euler scheme for singular kinetic SDEs driven by $α$-stable processes

TL;DR

The paper studies strong convergence of an Euler-type scheme for singular second-order SDEs driven by α-stable noise with degenerate coordinates. By leveraging regularization-by-noise techniques, a stochastic sewing framework, and sharp semigroup estimates in anisotropic Besov spaces, it proves existence/uniqueness of the solution and a strong convergence rate for the Euler approximation with rate n^{-(1/2+β/(α(1+α))∧1/2)+ε} for any ε>0, under anisotropic Hölder drift with β>1−α/2. An almost-sure convergence result is also established via a random dominating variable, with a rate matching the L_p bound up to ε. The analysis harmonizes second-order kinetic SDE behavior with first-order BDG-Lévy-type results, extending regularization-by-noise methods to this degenerate, jump-driven setting and providing rigorous guidance for numerical simulations of such systems.

Abstract

We study the strong approximation of the solutions to singular stochastic kinetic equations (also referred to as second-order SDEs) driven by -stable processes, using an Euler-type scheme inspired by [11]. For these equations, the stability index lies in the range , and the drift term exhibits anisotropic -Hölder continuity with . We establish a convergence rate of , which aligns with the results in [4] concerning first-order SDEs.

Paper Structure

This paper contains 13 sections, 16 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Notations
  4. Main results
  5. Auxiliary Tools
  6. Conditional shifted Stochastic Sewing Lemma and weighted John-Nirenberg Inequality
  7. Estimates on the semi-group
  8. A priori bounds
  9. Estimates of the functional
  10. Estimates of $S^{b,n,1}$, $S^{b,n,2}$
  11. Estimates of $S^{b,n,3}$
  12. Crucial conditional quadrature estimate
  13. Proof of the main result

Key Result

Proposition 2.3

For $s>0$, $q\in[1,\infty]$ and ${\boldsymbol{p}}\in[1,\infty]^2$, an equivalent norm of ${\mathbb B}^{s,q}_{{\boldsymbol{p}};{\boldsymbol{a}}}$ is given by where $[s]$ is the integer part of $s$. In particular, ${\mathbb C}^s_{{\boldsymbol{a}}}:=\mathbf{B}^{s,\infty}_{\infty;{\boldsymbol{a}}}$ is the anisotropic Hölder-Zygmund space, and for $s\in(0,1)$, there is a constant $C=C(\alpha,d,s)>0$ s

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.5
  • Corollary 2.6
  • Remark 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 22 more